(with the coefficients being real or complex numbers and an ≠ 0) is known by the fundamental theorem of algebra to have n (not necessarily distinct) complex roots x1, x2, ..., xn. Vieta's formulas relate the polynomial's coefficients { ak } to signed sums and products of its roots { xi } as follows:

Vieta's formulas are frequently used with polynomials with coefficients in any integral domainR. Then, the quotients ai/an{\displaystyle a_{i}/a_{n}} belong to the ring of fractions of R (or in R itself if an{\displaystyle a_{n}} is invertible in R) and the roots xi{\displaystyle x_{i}} are taken in an algebraically closed extension. Typically, R is the ring of the integers, the field of fractions is the field of the rational numbers and the algebraically closed field is the field of the complex numbers.

Vieta's formulas are then useful because they provide relations between the roots without having to compute them.

For polynomials over a commutative ring which is not an integral domain, Vieta's formulas are only valid when an{\displaystyle a_{n}} is a non-zerodivisor and P(x){\displaystyle P(x)} factors as an(x−x1)(x−x2)…(x−xn){\displaystyle a_{n}(x-x_{1})(x-x_{2})\dots (x-x_{n})}. For example, in the ring of the integers modulo 8, the polynomial P(x)=x2−1{\displaystyle P(x)=x^{2}-1} has four roots: 1, 3, 5, and 7. Vieta's formulas are not true if, say, x1=1{\displaystyle x_{1}=1} and x2=3{\displaystyle x_{2}=3}, because P(x)≠(x−1)(x−3){\displaystyle P(x)\neq (x-1)(x-3)}. However, P(x){\displaystyle P(x)} does factor as (x−1)(x−7){\displaystyle (x-1)(x-7)} and as (x−3)(x−5){\displaystyle (x-3)(x-5)}, and Vieta's formulas hold if we set either x1=1{\displaystyle x_{1}=1} and x2=7{\displaystyle x_{2}=7} or x1=3{\displaystyle x_{1}=3} and x2=5{\displaystyle x_{2}=5}.

(which is true since x1,x2,…,xn{\displaystyle x_{1},x_{2},\dots ,x_{n}} are all the roots of this polynomial), multiplying the factors on the right-hand side, and identifying the coefficients of each power of x.{\displaystyle x.}

Formally, if one expands (x−x1)(x−x2)⋯(x−xn),{\displaystyle (x-x_{1})(x-x_{2})\cdots (x-x_{n}),} the terms are precisely (−1)n−kx1b1⋯xnbnxk,{\displaystyle (-1)^{n-k}x_{1}^{b_{1}}\cdots x_{n}^{b_{n}}x^{k},} where bi{\displaystyle b_{i}} is either 0 or 1, accordingly as whether xi{\displaystyle x_{i}} is included in the product or not, and k is the number of xi{\displaystyle x_{i}} that are excluded, so the total number of factors in the product is n (counting xk{\displaystyle x^{k}} with multiplicity k) – as there are n binary choices (include xi{\displaystyle x_{i}} or x), there are 2n{\displaystyle 2^{n}} terms – geometrically, these can be understood as the vertices of a hypercube. Grouping these terms by degree yields the elementary symmetric polynomials in xi{\displaystyle x_{i}} – for xk, all distinct k-fold products of xi.{\displaystyle x_{i}.}

As reflected in the name, the formulas were discovered by the 16th century French mathematician François Viète, for the case of positive roots.

In the opinion of the 18th century British mathematician Charles Hutton, as quoted by Funkhouser,[1] the general principle (not only for positive real roots) was first understood by the 17th century French mathematician Albert Girard:

...[Girard was] the first person who understood the general doctrine of the formation of the coefficients of the powers from the sum of the roots and their products. He was the first who discovered the rules for summing the powers of the roots of any equation.